3.2070 \(\int \frac{(2+3 x) (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{15}{8} (1-2 x)^{5/2}-\frac{505}{24} (1-2 x)^{3/2}+\frac{1133}{8} \sqrt{1-2 x}+\frac{847}{8 \sqrt{1-2 x}} \]

[Out]

847/(8*Sqrt[1 - 2*x]) + (1133*Sqrt[1 - 2*x])/8 - (505*(1 - 2*x)^(3/2))/24 + (15*
(1 - 2*x)^(5/2))/8

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Rubi [A]  time = 0.0522967, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{15}{8} (1-2 x)^{5/2}-\frac{505}{24} (1-2 x)^{3/2}+\frac{1133}{8} \sqrt{1-2 x}+\frac{847}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]

[Out]

847/(8*Sqrt[1 - 2*x]) + (1133*Sqrt[1 - 2*x])/8 - (505*(1 - 2*x)^(3/2))/24 + (15*
(1 - 2*x)^(5/2))/8

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Rubi in Sympy [A]  time = 7.31127, size = 46, normalized size = 0.87 \[ \frac{15 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{505 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} + \frac{1133 \sqrt{- 2 x + 1}}{8} + \frac{847}{8 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)*(3+5*x)**2/(1-2*x)**(3/2),x)

[Out]

15*(-2*x + 1)**(5/2)/8 - 505*(-2*x + 1)**(3/2)/24 + 1133*sqrt(-2*x + 1)/8 + 847/
(8*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0362685, size = 32, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (45 x^3+185 x^2+631 x-685\right )}{6 x-3} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]

[Out]

(Sqrt[1 - 2*x]*(-685 + 631*x + 185*x^2 + 45*x^3))/(-3 + 6*x)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \[ -{\frac{45\,{x}^{3}+185\,{x}^{2}+631\,x-685}{3}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)*(3+5*x)^2/(1-2*x)^(3/2),x)

[Out]

-1/3*(45*x^3+185*x^2+631*x-685)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.34432, size = 50, normalized size = 0.94 \[ \frac{15}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{505}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1133}{8} \, \sqrt{-2 \, x + 1} + \frac{847}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)/(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

15/8*(-2*x + 1)^(5/2) - 505/24*(-2*x + 1)^(3/2) + 1133/8*sqrt(-2*x + 1) + 847/8/
sqrt(-2*x + 1)

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Fricas [A]  time = 0.227893, size = 32, normalized size = 0.6 \[ -\frac{45 \, x^{3} + 185 \, x^{2} + 631 \, x - 685}{3 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)/(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/3*(45*x^3 + 185*x^2 + 631*x - 685)/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right ) \left (5 x + 3\right )^{2}}{\left (- 2 x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)*(3+5*x)**2/(1-2*x)**(3/2),x)

[Out]

Integral((3*x + 2)*(5*x + 3)**2/(-2*x + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.213014, size = 59, normalized size = 1.11 \[ \frac{15}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{505}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1133}{8} \, \sqrt{-2 \, x + 1} + \frac{847}{8 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)/(-2*x + 1)^(3/2),x, algorithm="giac")

[Out]

15/8*(2*x - 1)^2*sqrt(-2*x + 1) - 505/24*(-2*x + 1)^(3/2) + 1133/8*sqrt(-2*x + 1
) + 847/8/sqrt(-2*x + 1)